Elliptic Curve Random Number Generation

ABSTRACT

An elliptic curve random number generator avoids escrow keys by choosing a point on the elliptic curve as verifiably random. An arbitrary string is chosen and a hash of that string computed. The hash is then converted to a field element of the desired field, the field element regarded as the x-coordinate of a point on the elliptic curve and the x-coordinate is tested for validity on the desired elliptic curve. If valid, the x-coordinate is decompressed to the point , wherein the choice of which is the two points is also derived from the hash value. Intentional use of escrow keys can provide for back up functionality. The relationship between P and is used as an escrow key and stored by for a security domain. The administrator logs the output of the generator to reconstruct the random number with the escrow key.

This application claims priority from U.S. Provisional Patent Application No. 60/644,982 filed on Jan. 21, 2005.

FIELD OF THE INVENTION

The present invention relates to systems and methods for cryptographic random number generation.

DESCRIPTION OF THE PRIOR ART

Random numbers are utilised in many cryptographic operations to provide underlying security. In public key infrastructures, for example, the private key of a key pair is generated by a random number generator and the corresponding public key mathematically derived therefrom. A new key pair may be generated for each session and the randomness of the generator therefore is critical to the security of the cryptographic system.

To provide a secure source of random numbers, cryptographically secure pseudorandom bit generators have been developed in which the security of each generator relies on a presumed intractability of the underlying number-theoretical problem. The American National Standards Institute (ANSI) has set up an Accredited Standards Committee (ASC) X9 for the financial services industry, which is preparing a American National Standard (ANS) X9.82 for cryptographic random number generation (RNG). One of the RNG methods in the draft of X9.82, called Dual_EC_DRBG, uses elliptic curve cryptography (ECC) for its security. Dual_EC_DRBG will hereinafter be referred to as elliptic curve random number generation (ECRNG).

Elliptic curve cryptography relies on the intractability of the discrete log problem in cyclic subgroups of elliptic curve groups. An elliptic curve E is the set of points (x, y) that satisfy the defining equation of the elliptic curve. The defining equation is a cubic equation, and is non-singular. The coordinates x and y are elements of a field, which is a set of elements that can be added, subtracted and divided, with the exception of zero. Examples of fields include rational numbers and real numbers. There are also finite fields, which are the fields most often used in cryptography. An example of a finite field is the set of integers modulo a prime q.

Without the loss of generality, the defining equation of the elliptic curve can be in the Weierstrass form, which depends on the field of the coordinates. When the field F is integers modulo a prime q>3, then the Weierstrass equation takes the form y²=x³+ax+b, where a and b are elements of the field F.

The elliptic curve E includes the points (x, y) and one further point, namely the point O at infinity. The elliptic curve E also has a group structure, which means that the two points P and

on the curve can be added to form a third point P+

. The point O is the identity of the group, meaning P+O=O+P=P, for all points P. Addition is associative, so that P+(

+R)=(P+

)+R, and commutative, so that P+

=

+R, for all points P,

and R. Each point P has a negative point −P, such that P+(−P)=O. When the curve equation is the Weierstrass equation of the form y²=x³+ax+b, the negative of P=(x, y) is determined easily as −P=(x, y). The formula for adding points P and

in terms of their coordinates is only moderately complicated involving just a handful of field operations.

The ECRNG uses as input two elliptic curve points P and

that are fixed. These points are not assumed to be secret. Typically, P is the standard generator of the elliptic curve domain parameters, and

is some other point. In addition a secret seed is inserted into the ECRNG.

The ECRNG has a state, which may be considered to be an integer s. The state s is updated every time the ECRNG produces an output. The updated state is computed as u=z(sP), where z( ) is a function that converts an elliptic curve point to an integer. Generally, z consists of taking the x-coordinate of the point, and then converting the resulting field element to an integer. Thus u will typically be an integer derived from the x-coordinate of the point s

The output of the ECRNG is computed as follows: r=t(z(s

)), where t is a truncation function. Generally the truncation function removes the leftmost bits of its input. In the ECRNG, the number of bits truncated depends on the choice of elliptic curve, and typically may be in the range of 6 to 19 bits.

Although P and

are known, it is believed that the output r is random and cannot be predicted. Therefore successive values will have no relationship that can be exploited to obtain private keys and break the cryptographic functions. The applicant has recognised that anybody who knows an integer d such that

=dP, can deduce an integer e such that ed=1 mod n, where n is the order of G, and thereby have an integer e such that P=e

. Suppose U=sP and R=s

, which are the precursors to the updated state and the ECRNG output. With the integer e, one can compute U from R as U=eR. Therefore, the output r=t(z(R)), and possible values of R can be determined from r. The truncation function means that the truncated bits of R would have to be guessed. The z function means that only the x-coordinate is available, so that decompression would have to be applied to obtain the full point R. In the case of the ECRNG, there would be somewhere between about 2⁶=64 and 2¹⁹ (i.e. about half a million) possible points R which correspond to r, with the exact number depending on the curve and the specific value of r.

The full set of R values is easy to determine from r, and as noted above, determination of the correct value for R determines U=eR, if one knows e. The updated state is u=z(U), so it can be determined from the correct value of R. Therefore knowledge of r and e allows one to determine the next state to within a number of possibilities somewhere between 2⁶ and 2¹⁹. This uncertainty will invariably be eliminated once another output is observed, whether directly or indirectly through a one-way function.

Once the next state is determined, all future states of ECRNG can be determined because the ECRNG is a deterministic function. (at least unless additional random entropy is fed into the ECRNG state) All outputs of the ECRNG are determined from the determined states of the ECRNG. Therefore knowledge of r and e, allows one to determine all future outputs of the ECRNG.

It has therefore been identified by the applicant that this method potentially possesses a trapdoor, whereby standardizers or implementers of the algorithm may possess a piece of information with which they can use a single output and an instantiation of the RNG to determine all future states and output of the RNG, thereby completely compromising its security. It is therefore an object of the present invention to obviate or mitigate the above mentioned disadvantages.

SUMMARY OF THE INVENTION

In one aspect, the present invention provides a method for computing a verifiably random point

for use with another point P in an elliptic curve random number generator comprising computing a hash including the point P as an input, and deriving the point

from the hash.

In another aspect, the present invention provides a method for producing an elliptic curve random number comprising generating an output using an elliptic curve random number generator, and truncating the output to generate the random number.

In yet another aspect, the present invention provides a method for producing an elliptic curve random number comprising generating an output using an elliptic curve random number generator, and applying the output to a one-way function to generate the random number.

In yet another aspect, the present invention provides a method of backup functionality for an elliptic curve random number generator, the method comprising the steps of computing an escrow key e upon determination of a point

of the elliptic curve, whereby P=e

, P being another point of the elliptic curve; instituting an administrator, and having the administrator store the escrow key e; having members with an elliptic curve random number generator send to the administrator, an output r generated before an output value of the generator; the administrator logging the output r for future determination of the state of the generator.

BRIEF DESCRIPTION OF THE DRAWINGS

An embodiment of the invention will now be described by way of example only with reference to the appended drawings wherein:

FIG. 1 is a schematic representation of a cryptographic random number generation scheme.

FIG. 2 is a flow chart illustrating a selection process for choosing elliptic curve points.

FIG. 3 is a block diagram, similar to FIG. 1 showing a further embodiment

FIG. 4 is flow chart illustrating the process implemented by the apparatus of FIG. 3.

FIG. 5 is a block diagram showing a further embodiment.

FIG. 6 is a flow chart illustrating yet another embodiment of the process of FIG. 2.

FIG. 7 is schematic representation of an administrated cryptographic random number generation scheme.

FIG. 8 is a flow chart illustrating an escrow key selection process.

FIG. 9 is a flow chart illustrating a method for securely utilizing an escrow key.

DETAILED DESCRIPTION OF THE INVENTION

Referring therefore to FIG. 1, a cryptographic random number generator (ECRNG) 10 includes an arithmetic unit 12 for performing elliptic curve computations. The ECRNG also includes a secure register 14 to retain a state value s and has a pair of inputs 16, 18 to receive a pair of initialisation points P,

. The points P,

are elliptic curve points that are assumed to be known. An output 20 is provided for communication of the random integer to a cryptographic module 22. The initial contents of the register 14 are provided by a seed input S.

This input 16 representing the point P is in a first embodiment, selected from a known value published as suitable for such use.

The input 18 is obtained from the output of a one way function in the form of a hash function 24 typically a cryptographically secure hash function such as SHA1 or SHA2 that receives as inputs the point P. The function 24 operates upon an arbitrary bit string A to produce a hashed output 26. The output 26 is applied to arithmetic unit 12 for further processing to provide the input

.

In operation, the ECRNG receives a bit string as a seed, which is stored in the register 14. The seed is maintained secret and is selected to meet pre-established cryptographic criteria, such as randomness and Hamming weight, the criteria being chosen to suit the particular application.

In order to ensure that d is not likely to be known (e.g. such that P=d

, and ed=1 mod n); one or both of the inputs 16, 18 is chosen so as to be verifiably random. In the embodiment of FIG. 1,

is chosen in a way that is verifiably random by deriving it from the output of a hash-function 24 (preferably one-way) whose input includes the point P. As shown in FIG. 2 an arbitrary string A is selected at step 202, a hash H of A is computed at step 204 with P and optionally S as inputs to a hash-based function F_(H)( ), and the hash H is then converted by the arithmetic unit 12 to a field element X of a desired field F at step 206. P may be pre-computed or fixed, or may also be chosen to be a verifiably random chosen value. The field element X is regarded as the x-coordinate of

(thus a “compressed” representation of

). The x-coordinate is then tested for validity on the desired elliptic curve E at step 208, and whether or not X is valid, is determined at step 210. If valid, the x-coordinate provided by element X is decompressed to provide point

at step 212. The choice of which of two possible values of the y co-ordinate is generally derived from the hash value.

The points P and

are applied at respective inputs 16, 18 and the arithmetic unit 12 computes the point s

where s is the current value stored in the register 14. The arithmetic unit 12 converts the x-coordinate of the point (in this example point s

) to an integer and truncates the value to obtain r=t(z(s

)). The truncated value r is provided to the output 20.

The arithmetic unit 12 similarly computes a value to update the register 14 by computing sP, where s is the value of the register 14, and converting the x-coordinate of the point sP to an integer u. The integer u is stored in the register to replace s for the next iteration. {ditto above}

As noted above, the point P may also be verifiably random, but may also be an established or fixed value. Therefore, the embodiment of FIG. 1 may be applied or retrofitted to systems where certain base points (e.g. P) are already implemented in hardware. Typically, the base point P will be some already existing base point, such as those recommended in Federal information Processing Standard (FIDS) 186-2. In such cases, P is not chosen to be verifiably random.

In general, inclusion of the point P in the input to the hash function ensures that P was determined before

is determined, by virtue of the one-way property of the hash function and since

is derived from an already determined P. Because P was determined before

, it is clearly understood that P could not have been chosen as a multiple of

(e.g. where P=e

), and therefore finding d is generally as hard as solving a random case of the discrete logarithm problem.

Thus, having a seed value S provided and a hash-based function F( )provided, a verifier can determine that

=F(S,P), where P may or may not be verifiably random. Similarly, one could compute P=F(S,

) with the same effect, though it is presumed that this is not necessary given that the value of P in the early drafts of X9.82 were identical to the base points specified in FIDS 186-2.

The generation of

from a bit string as outlined above may be performed externally of the ECRNG 10, or, preferably, internally using the arithmetic unit 12. Where both P and

are required to be verifiably random, a second hash function 24 shown in ghosted outline in FIG. 1 is incorporated to generate the coordinate of point P from the bit string A. By providing a hash function for at least one of the inputs, a verifiably random input is obtained.

It will also be noted that the output generated is derived from the x coordinate of the point sP. Accordingly, the inputs 16, 18 may be the x coordinates of P and

and the corresponding values of sP and s

obtained by using Montgomery multiplication techniques thereby obviating the need for recovery of the y coordinates.

An alternative method for choosing

is to choose

in some canonical form, such that its bit representation contains some string that would be difficult to produce by generating

=dP for some known d and P for example a representation of a name. It will be appreciated that intermediate forms between this method and the preferred method may also exist, where

is partly canonical and partly derived verifiably at random. Such selection of

, whether verifiably random, canonical, or some intermediate, can be called verifiable.

Another alternative method for preventing a key escrow attack on the output of an ECRNG, shown in FIGS. 3 and 4 is to add a truncation function 28 to ECRNG 10 to truncate the ECRNG output to approximately half the length of a compressed elliptic curve point. Preferably, this operation is done in addition to the preferred method of FIGS. 1 and 2, however, it will be appreciated that it may be performed as a primary measure for preventing a key escrow attack. The benefit of truncation is that the list of R values associated with a single ECRNG output r is typically infeasible to search. For example, for a 160-bit elliptic curve group, the number of potential points R in the list is about 2⁸⁰ , and searching the list would be about as hard as solving the discrete logarithm problem. The cost of this method is that the ECRNG is made half as efficient, because the output length is effectively halved.

Yet another alternative method shown in FIGS. 5 and 6 comprises filtering the output of the ECRNG through another one-way function F_(H2), identified as 34, such as a hash function to generate a new output. Again, preferably, this operation is performed in addition to the preferred method shown in FIG. 2, however may be performed as a primary measure to prevent key escrow attacks. The extra hash is relatively cheap compared to the elliptic curve operations performed in the arithmetic unit 12, and does not significantly diminish the security of the ECRNG.

As discussed above, to effectively prevent the existence of escrow keys, a verifiably random

should be accompanied with either a verifiably random P or a pre-established P. A pre-established P may be a point P that has been widely publicized and accepted to have been selected before the notion of the ECRNG 12, which consequently means that P could not have been chosen as P=e

because

was not created at the time when P was established.

Whilst the above techniques ensure the security of the system using the ECRNG by “closing” the trap door, it is also possible to take advantage of the possible interdependence of P and

, namely where P=e

, through careful use of the existence of e.

In such a scenario, the value e may be regarded as an escrow key. If P and

are established in a security domain controlled by an administrator, and the entity who generates

for the domain does so with knowledge of e (or indirectly via knowledge of d). The administrator will have an escrow key for every ECRNG that follows that standard.

Escrow keys are known to have advantages in some contexts. They can provide a backup functionality. If a cryptographic key is lost, then data encrypted under that key is also lost. However, encryption keys are generally the output of random number generators. Therefore, if the ECRNG is used to generate the encryption key K, then it may be possible that the escrow key e can be used to recover the encryption key K. Escrow keys can provide other functionality, such as for use in a wiretap. In this case, trusted law enforcement agents may need to decrypt encrypted traffic of criminals, and to do this they may want to be able to use an escrow key to recover an encryption key.

FIG. 7 shows a domain 40 having a number of ECRNG's 10 each associated with a respective member of the domain 40. The domain 40 communicates with other domains 40 a, 40 b, 40 c through a network 42, such as the internet. Each ECRNG of a domain has a pair of identical inputs P,

. The domain 40 includes an administrator 44 who maintains in a secure manner an escrow key e.

The administrator 44 chooses the values of P and

such that he knows an escrow key e such that

=eP. Other members of the domain 40 use the values of P and

, thereby giving the administrator 44 an escrow key e that works for all the members of the organization.

This is most useful in its backup functionality for protecting against the loss of encryption keys. Escrow keys e could also be made member-specific so that each member has its own escrow e′ from points selected by the administrator 44.

As generally denoted as numeral 400 in FIG. 8, the administrator initially selects a point P which will generally be chosen as the standard generator P for the desired elliptic curve 402. The administrator then selects a value d and the point

will be determined as

=dP 404, for some random integer d of appropriate size. The escrow key e is computed as e=d⁻¹ mod n 406, where n is the order of the generator P and stored by the administrator.

The secure use of such an escrow key 34 e is generally denoted by numeral 500 and illustrated in FIG. 9. The administrator 44 is first instituted 502 and an escrow keys e would be chosen and stored 504 by the administrator 44.

In order for the escrow key to function with full effectiveness, the escrow administrator 44 needs direct access to an ECRNG output value r that was generated before the ECRNG output value k (i.e. 16) which is to be recovered. It is not sufficient to have indirect access to r via a one-way function or an encryption algorithm. A formalized way to achieve this is to have each member with an ECRNG 12 communicate with the administrator 44 as indicated at 46 in FIG. 7, and step 506 in FIG. 9. This may be most useful for encrypted file storage systems or encrypted email accounts. A more seamless method may be applied for cryptographic applications. For example, in the SSL and TLS protocols, which are used for securing web (HTTP) traffic, a client and server perform a handshake in which their first actions are to exchange random values sent in the clear.

Many other protocols exchange such random values, often called nonces. If the escrow administrator observes these nonces, and keeps a log of them 508, then later it may be able to determine the necessary r value. This allows the administrator to determine the subsequent state of the ECRNG 12 of the client or server 510 (whoever is a member of the domain), and thereby recover the subsequent ECRNG 12 values. In particular, for the client who generally generates a random pre-master secret from which is derived the encryption key for the SSL or TLS session, the escrow key may allow recovery of the session key. Recovery of the session key allows recovery of the whole SSL or TLS session.

If the session was logged, then it may be recovered. This does not compromise long-term private keys, just session keys obtained from the output of the ECRNG, which should alleviate any concern regarding general suspicions related to escrows.

Whilst escrow keys are also known to have disadvantages in other contexts, their control within specific security domains may alleviate some of those concerns. For example, with digital signatures for non-repudiation, it is crucial that nobody but the signer has the signing key, otherwise the signer may legitimately argue the repudiation of signatures. The existence of escrow keys means the some other entity has access to the signing key, which enables signers to argue that the escrow key was used to obtain their signing key and subsequently generate their signatures. However, where the domain is limited to a particular organisation or part of an organisation it may be sufficient that the organisation cannot repudiate the signature. Lost signing keys do not imply lost data, unlike encryption keys, so there is little need to backup signing keys.

Although the invention has been described with reference to certain specific embodiments, various modifications thereof will be apparent to those skilled in the art without departing from the spirit and scope of the invention as outlined in the claims appended hereto. 

1-19. (canceled)
 20. A computer-implemented method of establishing an escrow key for a security domain within a network, comprising: establishing a pair of points (P,

) as respective inputs to an elliptic curve random number generator (ECRNG), wherein P=e

, and e is a relationship between the pair of points (P,

); storing the relationship e as an escrow key with an administrator; and generating from the ECRNG a random number for use in cryptographic operations within the security domain.
 21. The computer-implemented method of claim 20, wherein the pair of points (P,

) is a pair of elliptic curve points.
 22. The computer-implemented method of claim 20, wherein one point of the pair of points (P,

) is obtained from an output of a one way function.
 23. The computer-implemented method of claim 22, wherein the one way function is a hash function.
 24. The computer-implemented method of claim 22, wherein the other point of the pair of points (P,

) is utilized as an input to the one way function.
 25. The computer-implemented method of claim 24, wherein the one point of the pair of points (P,

) is

, and the other point of the pair of points (P,

) is P.
 26. The computer-implemented method of claim 20, wherein generating from the ECRNG the random number comprises logging an output of the ECRNG to reconstruct the random number with the escrow key by the administrator.
 27. A device, comprising: a memory; and at least one processor communicatively coupled with the memory and configured to: establish a pair of points (P,

) as respective inputs to an elliptic curve random number generator (ECRNG), wherein P=e

, and e is a relationship between the pair of points (P,

); store the relationship e as an escrow key with an administrator; and generate from the ECRNG a random number for use in cryptographic operations within a security domain.
 28. The device of claim 27, wherein the pair of points (P,

) is a pair of elliptic curve points.
 29. The device of claim 27, wherein one point of the pair of points (P,

) is obtained from an output of a one way function.
 30. The device of claim 29, wherein the one way function is a hash function.
 31. The device of claim 29, wherein the other point of the pair of points (P,

) is utilized as an input to the one way function.
 32. The device of claim 31, wherein the one point of the pair of points (P,

) is

, and the other point of the pair of points (P,

) is P.
 33. The device of claim 27, wherein generating from the ECRNG the random number comprises logging an output of the ECRNG to reconstruct the random number with the escrow key by the administrator.
 34. A non-transitory computer readable medium storing instructions which, when executed, cause a computing device to perform operations comprising: establishing a pair of points (P,

) as respective inputs to an elliptic curve random number generator (ECRNG), wherein P=e

, and e is a relationship between the pair of points (P,

); storing the relationship e as an escrow key with an administrator; and generating from the ECRNG a random number for use in cryptographic operations within a security domain.
 35. The non-transitory computer readable medium of claim 34, wherein the pair of points (P,

) is a pair of elliptic curve points.
 36. The non-transitory computer readable medium of claim 34, wherein one point of the pair of points (P,

) is obtained from an output of a one way function.
 37. The non-transitory computer readable medium of claim 36, wherein the one way function is a hash function.
 38. The non-transitory computer readable medium of claim 36, wherein the other point of the pair of points (P,

) is utilized as an input to the one way function.
 39. The non-transitory computer readable medium of claim 38, wherein the one point of the pair of points (P,

) is

, and the other point of the pair of points (P,

) is P. 